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1、SolutionstotheReviewQuestionsattheEndofChapter41. Inthesamewayaswemakeassumptionsaboutthetruevalueofbetaandnottheestimatedvalues,wemakeassumptionsaboutthetrueunobservabledisturbancetermsratherthantheirestimatedcounterparts,theresiduals.Weknowtheexactvalueoftheresiduals,sincetheyaredefinedbyli=H一.Sow
2、edonotneedtomakeanyassumptionsabouttheresidualssincewealreadyknowtheirvalue.Wemakeassumptionsabouttheunobservableerrortermssinceitisalwaysthetruevalueofthepopulationdisturbancesthatwearereallyinterestedin,althoughweneveractuallyknowwhattheseare.2. Wewouldliketoseenopatternintheresidualplot!Ifthereis
3、apatternintheresidualplot,thisisanindicationthatthereisstillsomeaction”orvariabilityleftiny?thathasnotbeenexplainedbyourmodel.Thisindicatesthatpotentiallyitmaybepossibletoformabettermodel,perhapsusingadditionalorcompletelydifferentexplanatoryvariables,orbyusinglagsofeitherthedependentorofoneormoreof
4、theexplanatoryvariables.Recallthatthetwoplotsshownonpages157and159,wheretheresidualsfollowedacyclicalpattern,andwhentheyfollowedanalternatingpatternareusedasindicationsthattheresidualsarepositivelyandnegativelyautocorrelatedrespectively.Anotherproblemifthereisa,patternz,intheresidualsisthat,ifitdoes
5、indicatethepresenceofautocorrelation,thenthismaysuggestthatourstandarderrorestimatesforthecoefficientscouldbewrongandhenceanyinferenceswemakeaboutthecoefficientscouldbemisleading.3. Theratiosforthecoefficientsinthismodelaregiveninthethirdrowafterthestandarderrors.Theyarecalculatedbydividingtheindivi
6、dualcoefficientsbytheirstandarderrors.=0.638+0.402及L0.891胃=o.96灰？=o.89(0.436)(0.291)(0.763)f-ratios1.461.38-1.17Theproblemappearstobethattheregressionparametersareallindividuallyinsignificant(i.e.notsignificantlydifferentfromzero),althoughthevalueofR2anditsadjustedversionarebothveryhigh,sothatthereg
7、ressiontakenasawholeseemstoindicateagoodfit.Thislookslikeaclassicexampleofwhatwetermnearmulticollinearity.Thisiswheretheindividualregressorsareverycloselyrelated,sothatitbecomesdifficulttodisentangletheeffectofeachindividualvariableuponthedependentvariable.Thesolutiontonearmulticollinearitythatisusu
8、allysuggestedisthatsincetheproblemisreallyoneofinsufficientinformationinthesampletodetermineeachofthecoefficients,thenoneshouldgooutandgetmoredata.Inotherwords,weshouldswitchtoahigherfrequencyofdataforanalysis(e.g.weeklyinsteadofmonthly,monthlyinsteadofquarterlyetc.).Analternativeisalsotogetmoredata
9、byusingalongersampleperiod(i.e.onegoingfurtherbackintime),ortocombinethetwoindependentvariablesinaratio(e.g.xztW).Other;moreadhocmethodsfordealingwiththepossibleexistenceofnearmulticollinearitywerediscussedinChapter4:-Ignoreit:ifthemodelisotherwiseadequate,i.e.statisticallyandintermsofeachcoefficien
10、tbeingofaplausiblemagnitudeandhavinganappropriatesign.Sometimes,theexistenceofmulticollinearitydoesnotreducetheratiosonvariablesthatwouldhavebeensignificantwithoutthemulticollinearitysufficientlytomaketheminsignificantItisworthstatingthatthepresenceofnearmulticollinearitydoesnotaffecttheBLUEproperti
11、esoftheOLSestimator-i.e.itwillstillbeconsistent,unbiasedandefficientsincethepresenceofnearmulticollinearitydoesnotviolateanyoftheCLRMassumptions1-4.However,inthepresenceofnearmulticollinearity,itwillbehardtoobtainsmallstandarderrors.Thiswillnotmatteriftheaimofthemodel-buildingexerciseistoproducefore
12、castsfromtheestimatedmodel,sincetheforecastswillbeunaffectedbythepresenceofnearmulticollinearitysolongasthisrelationshipbetweentheexplanatoryvariablescontinuestoholdovertheforecastedsample.-Droponeofthecollinearvariables-sothattheproblemdisappears.However,thismaybeunacceptabletotheresearcheriftherew
13、erestrongaprioritheoreticalreasonsforincludingbothvariablesinthemodel.Also,iftheremovedvariablewasrelevantinthedatageneratingprocessforytanomittedvariablebiaswouldresult.-Transformthehighlycorrelatedvariablesintoaratioandincludeonlytheratioandnottheindividualvariablesintheregression.Again,thismaybeu
14、nacceptableiffinancialtheorysuggeststhatchangesinthedependentvariableshouldoccurfollowingchangesintheindividualexplanatoryvariables,andnotaratioofthem.4. (a)TheassumptionofKomoscedasticityisthatthevarianceoftheerrorsisconstantandfiniteovertime.Technically,wewrite(b) Thecoefficientestimateswouldstill
15、bethe“correctones(assumingthattheotherassumptionsrequiredtodemonstrateOLSoptimalityaresatisfied),buttheproblemwouldbethatthestandarderrorscouldbewrong.Henceifweweretryingtotesthypothesesaboutthetrueparametervalues,wecouldendupdrawingthewrongconclusions.Infact,forallofthevariablesexcepttheconstant,th
16、estandarderrorswouldtypicallybetoosmall,sothatwewouldenduprejectingthenullhypothesistoomanytimes.(c) Thereareanumberofwaystoproceedinpractice,including-UsingKeteroscedasticityrobuststandarderrorswhichcorrectfortheproblembyenlargingthestandarderrorsrelativetowhattheywouldhavebeenforthesituationwheret
17、heerrorvarianceispositivelyrelatedtooneoftheexplanatoryvariables.-Transformingthedataintologs,whichhastheeffectofreducingtheeffectoflargeerrorsrelativetosmallones.5.(a)ThisiswherethereisarelationshipbetweenthehandTthresiduals.RecallthatoneoftheassumptionsoftheCLRMwasthatsucharelationshipdidnotexist.
18、Wewantourresidualstoberandom,andifthereisevidenceofautocorrelationintheresiduals,thenitimpliesthatwecouldpredictthesignofthenextresidualandgettherightanswermorethanhalfthetimeonaverage!(b) TheDurbinWatsontestisatestforfirstorderautocorrelation.Thetestiscalculatedasfollows.Youwouldrunwhateverregressi
19、onyouwereinterestedin,andobtaintheresiduals.Thencalculatethestatistic(-J2DW=2=22r-2YouwouldthenneedtolookupthetwocriticalvaluesfromtheDurbinWatsontables,andthesewoulddependonhowmanyvariablesandhowmanyobservationsandhowmanyregressors(excludingtheconstantthistime)youhadinthemodel.Therejection/non-reje
20、ctionrulewouldbegivenbyselectingtheappropriateregionfromthefollowingdiagram:Reject：positiveInconclusiveautocorrelationIIIDonotrejectRejectH:NOevidenceInconclusivenegativeofautocorrelationautocorrelationIIIIodLdu24-du4-dL4(c) Wehave60observations,andthenumberofregressorsexcludingtheconstanttermis3.Th
21、eappropriatelowerandupperlimitsare1.48and1.69respectively,sotheDurbinWatsonislowerthanthelowerlimit.Itisthusclearthatwerejectthenullhypothesisofnoautocorrelation.Soitlooksliketheresidualsarepositivelyautocorrelated.(d) a=四+Bax+故3,+BaZ+w,Theproblemwithamodelentirelyinfirstdifferences,isthatoncewecalc
22、ulatethelongrunsolution,allthefirstdifferencetermsdropout(asinthelongrunweassumethatthevaluesofallvariableshaveconvergedontheirownlongrunvaluessothatyt=yt-etc.)Thuswhenwetrytocalculatethelongrunsolutiontothismodel,wecannotdoitbecausethereisn,talongrunsolutiontothismodel!(e) Ayr=AI+0axli+Psx2t-+3-+匕T
23、heanswerisyes,thereisnoreasonwhywecannotuseDurbinWatsoninthiscase.Youmayhavesaidnoherebecausetherearelaggedvaluesoftheregressors(thexvariables)variablesintheregression.InfactthiswouldbewrongsincetherearenolagsoftheDEPENDENT引variableandhenceDWcanstillbeused.6. Ayr=+Qx*+B4n+A-1Ar2r-1+Bs%+A-4%Themajors
24、tepsinvolvedincalculatingthelongrunsolutionareto-setthedisturbancetermequaltoitsexpectedvalueofzero-dropthetimesubscripts-removealldifferencetermsaltogethersincethesewillallbezerobythedefinitionofthelongruninthiscontext.Followingthesesteps,weobtain=A+Ay+52+Py+P3Wenowwanttorearrangethistohaveallthete
25、rmsinxztogetherandsothatyisthesubjectoftheformula:iys2ft3P3Ay=-52-(APi)v_BBSX(A+A)yx9x1BaBJAThelastequationaboveisthelongrunsolution.7. Ramsey1sRESETtestisatestofwhetherthefunctionalformoftheregressionisappropriate.Inotherwords,wetestwhethertherelationshipbetweenthedependentvariableandtheindependent
26、variablesreallyshouldbelinearorwhetheranon-linearformwouldbemoreappropriate.Thetestworksbyaddingpowersofthefittedvaluesfromtheregressionintoasecondregression.Iftheappropriatemodelwasalinearone,thenthepowersofthefittedvalueswouldnotbesignificantinthissecondregression.IfwefailRamsey,sRESETtest,thenthe
27、easiest“solution“isprobablytotransformallofthevariablesintologarithms.Thishastheeffectofturningamultiplicativemodelintoanadditiveone.Ifthisstillfails,thenwereallyhavetoadmitthattherelationshipbetweenthedependentvariableandtheindependentvariableswasprobablynotlinearafterallsothatwehavetoeitherestimat
28、eanon-linearmodelforthedata(whichisbeyondthescopeofthiscourse)orwehavetogobacktothedrawingboardandrunadifferentregressioncontainingdifferentvariables.8. (a)Itisimportanttonotethatwedidnotneedtoassumenormalityinordertoderivethesampleestimatesofaandorincalculatingtheirstandarderrors.Weneededthenormali
29、tyassumptionatthelaterstagewhenwecometotesthypothesesabouttheregressioncoefficients,eithersinglyorjointly,sothattheteststatisticswecalculatewouldindeedhavethedistribution(forF)thatwesaidtheywould.(b)Onesolutionwouldbetouseatechniqueforestimationandinferencewhichdidnotrequirenormality.Butthesetechniq
30、uesareoftenhighlycomplexandalsotheirpropertiesarenotsowellunderstood,sowedonotknowwithsuchcertaintyhowwellthemethodswillperformindifferentcircumstances.Onepragmaticapproachtofailingthenormalitytestistoplottheestimatedresidualsofthemodel,andlookforoneormoreveryextremeoutliers.Thesewouldberesidualstha
31、taremuch“bigger”(eitherverybigandpositive,orverybigandnegative)thantherest.Itis,fortunatelyforus,oftenthecasethatoneortwoveryextremeoutlierswillcauseaviolationofthenormalityassumption.Thereasonthatoneortwoextremeoutlierscancauseaviolationofthenormalityassumptionisthattheywouldleadthe(absolutevalueof
32、the)skewnessand/orkurtosisestimatestobeverylarge.Oncewespotafewextremeresiduals,weshouldlookatthedateswhentheseoutliersoccurred.Ifwehaveagoodtheoreticalreasonfordoingso,wecanaddinseparatedummyvariablesforbigoutlierscausedby,forexample,wars,changesofgovernment,stockmarketcrashes,changesinmarketmicros
33、tructure(e.g.thebigbang”of1986).Theeffectofthedummyvariableisexactlythesameasifwehadremovedtheobservationfromthesamplealtogetherandestimatedtheregressionontheremainder.Ifweonlyremoveobservationsinthisway,thenwemakesurethatwedonotloseanyusefulpiecesofinformationrepresentedbysamplepoints.9. (a)Paramet
34、erstructuralstabilityreferstowhetherthecoefficientestimatesforaregressionequationarestableovertime.Iftheregressionisnotstructurallystable,itimpliesthatthecoefficientestimateswouldbedifferentforsomesubsamplesofthedatacomparedtoothers.Thisisclearlynotwhatwewanttofindsincewhenweestimatearegression,wear
35、eimplicitlyassumingthattheregressionparametersareconstantovertheentiresampleperiodunderconsideration.(b)1981M1-1995M12rt = 0.0215 + 1.491 rmt?SS=O.189T=1801981M1-1987M1Ort = 0.0163 + 1.308 rmt/?SS= 0.079T=821987M11-1995M12rt = 0.0360 + 1.613 rmt/?SS=0.082 T=98(c) Ifwedefinethecoefficientestimatesfor
36、thefirstandsecondhalvesofthesampleasaand,anda2and2respectively,thenthenullandalternativehypothesesareH0:=sandandHi:a2or2(d) TheteststatisticiscalculatedasTeststat.=RSS-(RSS.+RSS,)(T-2&)0.189-(0.079+0.()82)180-4”，RSSl+RSS2k0.079+0.0822ThisfollowsanFdistributionwith(k,T-2kdegreesoffreedom.尺2,176)=3.05
37、atthe5%level.Clearlywerejectthenullhypothesisthatthecoefficientsareequalinthetwosub-periods.10. Thedatawehaveare1981M1-1995M12 : 0.0215 + 1.491 Rmt/?SS=0.189=1801981M1-1994M12rt = 0.0212 + 1.478 Rmt?SS=O.148上1681982M1-1995M12rt = 0.0217 + 1.523 Rmt?SS=O.182=168First,theforwardpredictivefailuretest-i
38、.e.wearetryingtoseeifthemodelfor1981M1-1994M12canpredict1995M1-1995M12.Theteststatisticisgivenby型g*X =好3 W = 3.832RSSl0.14812Where71isthenumberofobservationsinthefirstperiod(i.e.theperiodthatweactuallyestimatethemodelover),andTiisthenumberofobservationswearetryingtopredict.Theteststatisticfollowsanf
39、distributionwith(Ti,Ti-Qdegreesoffreedom.尺12,166)=1.81atthe5%level.Sowerejectthenullhypothesisthatthemodelcanpredicttheobservationsfor1995.Wewouldconcludethatourmodelisnouseforpredictingthisperiod,andfromapracticalpointofview,wewouldhavetoconsiderwhetherthisfailureisaresultofa-typicalbehaviourofthes
40、eriesout-of-sample(i.e.during1995),orwhetheritresultsfromagenuinedeficiencyinthemodel.Thebackwardpredictivefailuretestisalittlemoredifficulttounderstand,althoughnomoredifficulttoimplementTheteststatisticisgivenbyRSS-RSSi.Ti-k0.189-0.182.168-2n_*=*=().)32RSS、T20.18212Nowweneedtobealittlecarefulinouri
41、nterpretationofwhatexactlyarethe“firstandsecondsampleperiods.Itwouldbepossibletodefine71asalwaysbeingthefirstsampleperiod.ButIthinkiteasiertosaythatTiisalwaysthesampleoverwhichweestimatethemodel(eventhoughitnowcomesafterthehold-out-sample).ThusTiisstillthesamplethatwearetryingtopredict,eventhoughitc
42、omesfirst.Youcanuseeithernotation,butyouneedtobeclearandconsistent.IfyouwantedtochoosetheotherwaytotheoneIsuggest,thenyouwouldneedtochangethesubscript1everywhereintheformulaabovesothatitwas2,andchangeevery2sothatitwasa1.Eitherway,weconcludethatthereislittleevidenceagainstthenullhypothesis.Thusourmod
43、elisabletoadequatelyback-castthefirst12observationsofthesample.11. Bydefinition,variableshavingassociatedparametersthatarenotsignificantlydifferentfromzeroarenot,fromastatisticalperspective,helpingtoexplainvariationsinthedependentvariableaboutitsmeanvalue.Onecouldthereforearguethatempirically,theyse
44、rvenopurposeinthefittedregressionmodel.Butleavingsuchvariablesinthemodelwilluseupvaluabledegreesoffreedom,implyingthatthestandarderrorsonalloftheotherparametersintheregressionmodel,willbeunnecessarilyhigherasaresult.Ifthenumberofdegreesoffreedomisrelativelysmall,thensavingacouplebydeletingtwovariabl
45、eswithinsignificantparameterscouldbeuseful.Ontheotherhand,ifthenumberofdegreesoffreedomisalreadyverylarge,theimpactoftheseadditionalirrelevantvariablesontheothersislikelytobeinconsequential.12. Anoutlierdummyvariablewilltakethevalueoneforoneobservationinthesampleandzeroforallothers.TheChowtestinvolv
46、essplittingthesampleintotwoparts.Ifwethentrytoruntheregressiononboththesub-partsbutthemodelcontainssuchanoutlierdummy,thentheobservationsonthatdummywillbezeroeverywhereforoneoftheregressions.Forthatsub-sample,theoutlierdummywouldshowperfectmulticollinearitywiththeinterceptandthereforethemodelcouldnotbeestimated.
